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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Nalyd on February 04, 2014, 11:07:54 pm

Title: Help with trigonometric identities
Post by: Nalyd on February 04, 2014, 11:07:54 pm: Hey everyone. To start this off, I'd like to point out I am currently undergoing year 12 SACE and having some issues with simplifying trigonometric identities in Specialist Maths. I know it may seem weird I'm posting here for advice, but it seems to be the most active place I can find, filled with people who know what they're doing. The VCE might not even cover this sort of stuff, I'm not sure, but if you can help me I'd greatly appreciate it.

So, I'm looking for some general tips to be able to solve and work my way through these problems. One of the questions I came across tonight was:

Prove that:

$\cos 3\theta = 4\cos ^3 \theta - 3\cos \theta$

With the worked solution of:

$\cos 3\theta$
$=\cos(2\theta + \theta)$
$=\cos2\theta \cos\theta - \sin2\theta \sin\theta$
$=(2\cos ^2 \theta - 1)\cos \theta - 2\sin \theta \cos \theta \sin \theta$
$=2\cos ^3 \theta - \cos \theta - 2\cos \theta (\sin ^2 \theta)$
$=2\cos ^3 \theta - \cos \theta - 2\cos \theta (1-\cos ^2 \theta)$
$=2\cos ^3 \theta - \cos \theta - 2\cos \theta + 2\cos ^3 \theta$
$=4\cos ^3 \theta - 3\cos \theta$

I was able to use the addition formula and substitute in the duplication formula fine, but I have no idea how I'm supposed to know what to factor out where to help with my solving after I've done that. If you have any good tips or advice on this sort of stuff, please send it my way. It seems like every third trigonometric identities question I get, I can't solve, and becoming really stressful.

If you can't help, thanks for taking the time to read this, and sorry :P
Title: Re: Help with trigonometric identities
Post by: b^3 on February 04, 2014, 11:21:57 pm: Sometimes you just need to play around with what you have, or look for other trigonometric identities hidden inside (so really recognising when a certain groups of terms pops out and using it to turn it into something more useful). If you get really stuck then expand the whole expression (provided it's not too lengthy), and from that try and say group cos's and see if something else pops out (such as the question above). It really just depends on the question, some are approached in different ways, but being able to recognise when there is an equivalent expression that will be of more use to you will help a lot. So yeah, make sure you know your double and compound angle formulas I guess, and practice trying to recognise them.

I have a way I remember the compound angle formulas for sine and cosine.

If it's cosine, then you have cos cos sin sin, with the sign between the cos's and sine's being opposite to whats inside the bracket. So think "Same trig, different sign".
e.g.
$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$
$\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$
For sine's, it's cos sin sin cos, with the same sign as whats in the bracket.
e.g.
Think "alternating trig, same sign".
$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$
$\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$

I'm not sure if I really told you anything that you didn't already know, but hope it helps. (It might be a bit
Title: Re: Help with trigonometric identities
Post by: Nalyd on February 05, 2014, 08:36:33 am: Thanks b^3 for the reply! I guess I'll just having to keep practicing them and try and learn to recognise what to do and where. I think I need to start trying to factorise more when solving them, as often the ones I can't do, the solutions have done so.

I know all the formulas fine, but that's a pretty cool way to remember it! Will come in handy if it ever slips my mind (it does sometimes). Thanks again, I appreciate it. :)